You need to look at the documentation for the GeoProcessor.
Basically, you will create a new geoprocessor object like this:
gp = new esri.tasks.Geoprocessor("http://YourServer/ArcGIS/rest/services/foo/GPServer/bar");
Then you will need to create a parameter object, with the two required parameters. for this, I'm assuming you will need to take the user's input and put in the zip code, as the RTE_box if required.
so your code can be somethinglike this:
var zipcode=<input from user>;
var type=<RTE input from user>;
var params={"ZIP":zipcode,"Rte_Box":type };
You will then call the execute function on the geoprocessor, also passing in the callback function, like this:
//your callback function is called onResults
gp.execute(params, onResults);
In your onResults function, you will get the resulting featureset, which you can then draw on the map as graphics.
There are several samples available, like this one: Service area task
Best Answer
If this does not need to be automated then, since you are using ArcMap, you could use the built-in Create Thumbnails function for a layer.
Alternatively you could make a simple map showing only your shapefile, save it as a JPEG and down-sample it to thumbnail size with Photoshop or GIMP (you can do this in Arc obviously and set your resolution and page size way down)
If you have ArcMap, you don't need Mapnik or any other open source tools just to create an image of your data and you only need Python or Java etc if you want to automate the process!